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A030200 Expansion of q^(-1/2) * eta(q) * eta(q^11) in powers of q. 3
1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 0, 0, 2, 1, 0, 2, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 2, 0, 0, 0, 1, -1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, -1, 0, -2, 0, 0, -1, 0, 0, 0, 1, -1, -2, 0, 2, 1, 0, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 2, 1, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,24

COMMENTS

Number 52 of the 74 eta-quotients listed in Table I of Martin 1996.

REFERENCES

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 203. MR1471703 (98g:14032)

LINKS

Table of n, a(n) for n=0..104.

FORMULA

Euler transform of period 11 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, ...]. - Michael Somos, Nov 20 2006

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(11^e) = 1, b(p^e) = (e-1)%3 - 1 if f=0, b(p^e) = e+1 if f=3, b(p^e) = (1 + (-1)^e) / 2 if f=1 where f = number of zeros of x^3 - x^2 - x - 1 modulo p. - Michael Somos, Nov 20 2006

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(11*k)).

Sum over all solutions to x^2 + x*y + 3*y^2 = 2*n + 1 with x>0 odd of (-1)^y. - Michael Somos, Jan 29 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 pi i t).

EXAMPLE

1 - x - x^2 + x^5 + x^7 - x^11 + x^13 - x^15 - x^16 - x^18 + 2*x^23 + ...

q - q^3 - q^5 + q^11 + q^15 - q^23 + q^27 - q^31 - q^33 - q^37 + 2*q^47 +...

PROG

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; qfrep( [1, 0; 0, 11], n)[n] - qfrep( [3, 1; 1, 4], n)[n])} /* Michael Somos, Nov 20 2006 */

(PARI) {a(n) = local(A, p, e, f); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==11, 1, f = sum( k=0, p-1, (k^3 - k^2 -k - 1)%p == 0); if( f==0, (e-1)%3-1, if( f==1, (1 + (-1)^e) / 2, e+1)))))))} /* Michael Somos, Nov 20 2006 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^11 + A), n))} /* Michael Somos, Nov 20 2006 */

CROSSREFS

Cf. A106276. Convolution square is A006571.

Sequence in context: A138514 A143540 A208664 * A095734 A137269 A112201

Adjacent sequences:  A030197 A030198 A030199 * A030201 A030202 A030203

KEYWORD

sign

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 26 01:20 EDT 2014. Contains 248566 sequences.